In this paper, we review
recent studies on the magnetic helicity changes of solar active regions
by photospheric horizontal motions.
Recently, Chae(2001) developed a methodology to determine
the magnetic helicity change rate via photospheric horizontal motions.
We have applied this methodology to four cases:
(1) NOAA AR 8100 which has a series of homologous X-ray flares,
(2) three active regions which have four eruptive major X-ray flares,
(3) NOAA AR 9236 which has three eruptive X-class flares,
and (4) NOAA AR 8668 in which a large filament was under formation.
As a result, we have found several interesting results.
First, the rate of magnetic helicity injection strongly depends
on an active region and its evolution. Its mean rate ranges from
4 to 17 imes 1040 mbox{ Mx}^2 mbox{ h}-1.
Especially when the homologous flares occurred and
when the filament was formed, significant rates of magnetic
helicity were continuously deposited
in the corona via photospheric shear flows.
Second, there is a strong positive correlation between the magnetic
helicity accumulated
during the flaring time interval of the homologous flares in AR 8100
and the GOES X-ray flux integrated over the flaring time.
This indicates that the occurrence of a series of homologous
flares is physically related to the accumulation of magnetic
helicity in the corona by photospheric shearing motions. Third,
impulsive helicity variations took place near the flaring times of
some strong flares. These impulsive variations whose time
scales are less than one hour are
attributed to localized velocity kernels around the polarity inversion line.
Fourth, considering the filament eruption associated with an X1.8 flare
started about 10 minutes before the impulsive variation of the helicity change rate,
we suggest that the impulsive helicity variation
is not a cause of the eruptive solar flare but its result.
Finally, we discuss the physical implications
on these results and our future plans.

We address the question whether there exist sympathetic
coronal mass ejections (CMEs), which take place
almost simultaneously in different locations with
a certain physical connection.
For this study, the following three investigations are performed.
First, we have examined the waiting-time distribution
of the CMEs that were observed by {it SOHO}/LASCO
from February 1999 to December 2001.
The observed waiting time distribution is found to
be well approximated by a time-dependent Poisson distribution
without any noticeable overabundance at short waiting times.
Second, we have investigated the angular difference distribution
of successive CME pairs to examine their spatial correlations.
A remarkable overabundance relative to background levels is found within 10arcdeg
of the
position angle difference, which supports the existence of
quasi-homologous CMEs that sequentially occur in the same active region.
The above results both indicate that sympathetic (interdependent) CMEs
are far less frequent than independent CMEs.
Third, we have examined the EIT running difference images and
the LASCO images of quasi-simultaneous CME pairs, and found
a candidate of
the sympathetic CME pair, of which the second CME may be
initiated by eruption of the first CME.
Possible mechanisms of the sympathetic CME triggering are discussed.

In this paper,
we investigate impulsive variations of magnetic helicity change rate
associated with eruptive solar flares (three X-class flares and one
M-Class flare) accompanying halo CMEs. By analyzing
four sets of 1 minute cadence full-disk magnetograms
taken by Michelson Doppler Imager (MDI) on board Solar and
Heliospheric Observatory (SOHO),
we have determined the rates of magnetic helicity transport
due to horizontal photospheric motions.
We have found that
magnetic helicity of the order of 1041 Mx{^2}
was impulsively injected into the corona
around the flaring peak time of all the flares.
We also found that
there is a positive correlation between
the impulsively injected magnetic helicity and
the X-ray peak flux of the associated flare.
The impulsive helicity variations are attributed to
horizontal velocity kernels
localized near the polarity inversion lines.
Finally, we report that there is a close spatial proximity between
the horizontal velocity kernels and Hα bright points.

A comprehensive statistical study is performed to address the question
whether two classes of coronal mass ejections (CMEs) exist.
A total of 3217 CME events observed
by SOHO/LASCO in 1996 to 2000 have been analyzed.
We have examined the distributions of CMEs according
to speed and acceleration, respectively, and
investigated the correlation
between speed and acceleration of CMEs. This
statistical analysis is conducted for two
subsets containing those CMEs which show
a temporal and spatial association
either with GOES X-ray solar flares or with
eruptive filaments.
%The number of CMEs in deceleration is comparable to the
%number of CMEs in acceleration.
We have found that
CMEs associated with flares have a higher median
speed than those associated with eruptive filaments and that
the median speed of CMEs
associated with strong flares is higher than that
of weak-flare-associated CMEs.
The distribution of CME acceleration shows a conspicuous
peak near zero, not only for the whole data set, but also
for the two subsets associated either with
solar flares or with eruptive filaments.
However, we have confirmed that
the CMEs associated with major flares tend to be more decelerated
than the CMEs related to eruptive filaments.
The fraction of flare-associated CMEs has a tendency to increase with the CME speed,
whereas the fraction of eruptive-filament-associated CMEs tends
to decrease
with the CME speed.
This result supports the concept of two CME classes.
We have found a possibility of two components in the CME speed distribution
for both the CME data associated with flares larger than M1 class
and the CME data related with limb flares.
Our results suggest that the apparent single-peak distribution
of CME speed can be attributed to the projection effect and
possibly to abundance of small flares too.
We also note that there is a possible correlation
between the speed of CMEs and the time-integrated X-ray flux
of the CME-associated limb flares.

A statistical study is performed on X-ray flares stronger than C1 class that erupted
during the solar maximum between 1989 and 1991. We have investigated the flaring
time interval distribution (waiting-time distribution) and the spatial correlation of
successive flare pairs. The observed waiting-time distribution for the whole data is
found to be well represented by a nonstationary Poisson probability function with time
varying mean flaring rates.
The period most suitable for a constant mean flaring rate is determined
to be 2-3 days by a Kolmogorov-Smirnov test.
We have also found that the waiting-time distribution for flares in
individual active regions follows a stationary Poisson probability function
mexp(-mt) with a corresponding mean flaring
rate.
Therefore, the flaring probability within a given time is given by 1-exp(-mt), when
the mean flaring rate m is properly estimated.
It is also found that there is no systematic relationships between peak fluxes of
flares and their waiting-time distributions.
The above findings support the idea that the
solar corona is in a self-organized critical state.
A comparison of the angular distances of successively observed flare pairs with those of
hypothetical flare pairs generated by random distribution shows a positive angular
correlation within about 10 degrees (sim 180 arcsec in the observing field)
of angular separation, which suggests
that homologous flares occurring in the same active region should outnumber
sympathetic flares.

It is widely believed that solar magnetic fields are force-free
in the solar corona, but not in the solar photosphere at all.
In order to examine the force-freeness of active region magnetic fields
at the photospheric level, we have calculated the integrated magnetic forces
for 12 vector magnetograms of three
flare-productive active regions. The magnetic field vectors are
derived from
simultaneous Stokes profiles of the Fe I doublet 6301.5 and 6302.5
obtained
by the Haleakala Stokes Polarimeter of Mees Solar Observatory,
with a non-linear least square method adopted for field calibration.
The resulting vertical Lorentz force normalized to the total magnetic pressure force
|F_z/F_p| ranges from 0.06 to 0.36 with a median value of
0.13, which is smaller than the values (sim 0.4)
obtained by Metcalf et al.
who applied a weak field derivative method to the Stokes profiles of Na I 5896.
Our results indicate that the photospheric
magnetic fields are not so far from force-free as conventionally regarded.
As a good example of a linear force-free field, AR 5747 is
examined. By applying three different methods (a most probable value method,
a least square fitting method, and comparison with linear force-free
solutions), we have derived
relatively consistent linear force-free coefficients for AR 5747.
It is found that the scaled downward Lorentz force
(|F_z/F_p|) in the solar photosphere
decreases with increasing α .
Our results also show that the
force-freeness of photospheric
magnetic fields depends not only on the character of the active region,
but also on
its evolutionary status.

We have examined a possibility
for improvement of the STOA (Shock Time Of Arrival) model
for interplanetary shock propagation.
In the STOA model,
the shock propagating velocity is given by V_s sim R-N
with N=0.5,
where R is the heliocentric distance.
Noting observational and numerical findings that
the radial dependence of shock wave velocity depends on
initial shock wave velocity, we suggest a simple modified
STOA model (STOA-2)
which has a linear relationship between initial
coronal shock wave velocity
(Vis)
and its deceleration exponent(N),
N=0.05+4 imes10-4Vis, where Vis is a numeric value expressed
in units of km s-1.
Our results show that the STOA-2 model not only removes
a systematic dependence of the transit time difference
predicted by the previous
STOA model on initial shock velocity,
but also reduces the number of events with large
transit time differences.

Sympathetic flares are
a pair of flares which occur almost simultaneously
in different active regions, not by chance, but
due to some
physical connections. In this paper,
statistical evidence for
the existence of sympathetic flares is presented.
From GOES X-ray flare data,
we have collected 48 pairs of near simultaneous flares
whose positional information
and Yohkoh/SXT images are available.
To select the active regions which probably have sympathetic flares,
we have estimated the ratio R of actual flaring overlap time to
random-coincidence overlap time for 38 active region pairs.
We have then compared the waiting-time distributions
for the two different groups of active region pairs (R>1 and R<1)
with
corresponding nonstationary Poisson distributions.
As a result, we find a remarkable overabundance in
short waiting times
for the group with R>1. This is
the first time such strong statistical
evidence has been found for the existence of sympathetic flares.
To examine the role of interconnecting
coronal loops, we have also conducted the same analysis for two
subgroups of the R>1 group: one
with interconnecting
X-ray loops and the other without.
We do not find any statistical evidence
that the subgroup with interconnecting
coronal loops is more
likely to produce sympathetic flares
than the subgroup without.
For the subgroup with loops, we find that
sympathetic flares favor active region pairs with
transequatorial loops.

We present observational
evidence that the occurrence of homologous flares
in an active region is physically related to
the injection of magnetic helicity by
horizontal photospheric motions. We have analyzed
a set of 1 minute cadence magnetograms of NOAA AR 8100
taken over a period of 6.5 hours by Michelson Doppler Imager (MDI)
on board Solar and Heliospheric Observatory (SOHO).
During this observing time span,
seven homologous flares took place in the active region.
We have computed the magnetic helicity injection rate into
the solar atmosphere by photospheric shearing motions, and
found that a significant amount of magnetic helicity was
injected during the observing period.
In a strong M4.1 flare,
the magnetic helicity injection rate impulsively increased
and peaked at the same time as the X-ray flux did.
The flare X-ray flux integrated over the X-ray emission
time strongly correlates with the magnetic
helicity injected during the flaring interval.
The integrated X-ray flux is found to be a
logarithmically increasing function of
the injected magnetic helicity.
Our results suggest that injection of helicity and
abrupt increase of helicity magnitude play a
significant role in flare triggering.